I can't answer all of your questions, but I can tell you what the ballast relay is/was for in older cars that I am familiar with. In older cars with an ignition coil and a distributor with points it is necessary to have 12 volts to the coil for starting and 9 volts to the coil for running. There are therefore 2 circuits to the coil from the battery, one of which incorporates a resistor (the ballast resistor) to reduce the voltage to 9 volts. The ballast relay switches between these 2 circuits. The relay is sometimes incorporated into the starter motor solenoid circuits when these are part of the starter.
If the coil is allowed to run on 12 volts it will sometimes overheat and the ignition system will not function and you have to sit for about 45 minutes before the car will run again (my experience).
I'm not familiar with the modern technology and so can't tell you if these circuits are still required.
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